Download Lectures 15 and 16 - NUS Physics Department

Ground Rules
PC1221 Fundamentals of
Physics I
„
Lectures 15 and 16
„
„
„
Potential Energy
„
„
Dr Tay Seng Chuan
„
„
Switch off your handphone and pager
Switch off your laptop computer and keep it
No talking while lecture is going on
No gossiping while the lecture is going on
Raise your hand if you have question to ask
Be on time for lecture
Be on time to come back from the recess break to
continue the lecture
Bring your lecturenotes to lecture
1
2
Conservative
Forces
Isolated System
An isolated system is one for which there
are no energy transfers across the
boundary. The energy in such a system is
conserved , i.e., at anytime the sum is a
constant but its form can change in part
or in whole. E.g., a block sliding across a
frictionless table is moving in an isolated
system. If there is friction on the table
(rough surface), the block is not sliding in
an isolated system any more.
Conservative forces have these two equivalent properties:
• The work done by a conservative force on a particle
moving between any two points is independent of the
path taken by the particle.
•
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The work done by a conservative force on a particle
moving through any closed path is zero.
The gravitational force (free fall) is one example of a
conservative force, and the force that a spring exerts on
any object attached to the spring is another example.
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Potential Energy
„
System Example
Potential energy is the energy
associated with the configuration of a
system of objects that exert forces on
each other. Eg, if you stand on top of
Kent Ridge Hill, your potential energy is
larger than what you have now in this
lecture theatre at a lower ground.
„
„
„
„
„
When conservative forces act within an
isolated system, the kinetic energy gained
(or lost) by the system as its members
change their relative positions is balanced
by an equal loss (or gain) in potential
energy.
„
„
This is Conservation of
Mechanical Energy.
Energy
This system consists of Earth and
a book
Do work on the system by lifting
the book through Δy
The work done is mgyb – mgya
The energy storage mechanism is
called potential energy
Potential energy is always
associated with a system of two or
more interacting objects
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6
Gravitational Potential Energy
„
„
The gravitational potential energy
depends only on the vertical height of
the object above Earth’s surface
In solving problems, you must choose
a reference configuration for which
the gravitational potential energy is
set equal to some reference value,
normally zero
„
The choice is arbitrary because you
normally need the difference in
potential energy, which is independent of
the choice of reference configuration.
Therefore the choice of reference frame
is not important. E.g., the distance
between a and b is always the same
regardless of its reference with respect
to X or Y.
Systems with Multiple Particles
a
„
b
„
„
X
We can extend our definition of a
system to include multiple objects
The force can be internal to the system
The kinetic energy of the system is the
algebraic sum of the kinetic energies of
the individual objects
Y
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8
Conservation of Mechanical
Energy
„
The mechanical energy of a system is the
algebraic sum of the kinetic (K ) and potential
energies (Ug) in the system
„
„
Conservation of Mechanical
Energy, example
Emech = K + Ug
The statement of Conservation of Mechanical
Energy for an isolated system (An isolated
system is one for which there are no energy
transfers across the boundary) is
Kf + Uf = Ki+ Ui
(i.e., final sum = initial sum)
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„
„
„
Gain in
kinetic
energy
Elastic Potential Energy
„
„
„
with a spring
The force the spring exerts (on a block,
for example) is Fs = - kx
The work done by an external applied
force on a spring-block system is
„
„
„
„
„
W = ½ kxf2 – ½ kxi2 (x is measured from
equilibrium position.)
The work is equal to the difference between
the initial and final values of an expression
related to the configuration of the system
This expression is the elastic potential
energy: Us = ½ kx2, where x is the
distance from the natural position.
The elastic potential energy can be
regarded as the energy stored in the
deformed spring
The stored potential energy can be
converted into kinetic energy
Loss in
potential
energy
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Elastic Potential Energy
Elastic Potential Energy is associated
„
Look at the work done
by the book as it falls
from some height to a
lower height
Won book = ΔKbook
Also, W = mgyb – mgya
So, ΔK = -ΔUg
„
The elastic potential energy (U )stored in a spring is
zero whenever the spring is not deformed (U = 0
when x = 0)
„
„
„
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The energy is stored in the spring only when the spring is
stretched or compressed
The elastic potential energy is a maximum when
the spring has reached its maximum extension or
compression
The elastic potential energy (Us = ½ kx2) is always
positive because x2 will always be positive
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Conservation of Energy,
Example 1 (Drop a Ball)
„
Initial conditions:
„
„
„
„
Example. Dave Johnson, the bronze medalist at the
1992 Olympic decathlon in Barcelona, leaves the ground
at the high jump with vertical velocity component 6.00
m/s. How far does his center of mass move up as he
makes the jump?
Ei = Ki + Ui = mgh
Answer:
The ball is dropped from rest, so Ki = 0
The configuration for zero potential
energy is the ground, i.e., when the
ball hits the ground its potential
energy will become 0
Conservation rules applied at some
point y above the ground gives
„
From leaving ground to the highest
point,
Ki+ U i = K f + U
(
)
1
2
m ( 6.00 m s) + 0 = 0 + m 9.80 m s2 y
2
∴y=
½ mvf2 + mgy = mgh
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Example. Three identical balls are thrown
from the top of building all with the same
initial speed. The first is thrown horizontally,
the second at some angle above the
horizontal, and the third at some angle
below the horizontal. Neglecting air
resistance, rank the speeds of the balls at
the instant each hits the ground
You might rank ball-2 with the highest speed because it travels to a
higher point before it starts to fall. As the height will be higher so the
speed when ball-2 hits the ground will be larger.
In actual case, all the 3 speeds will be the same when the balls hit the
ground. This is because the initial positions of the balls are the same (so
the potential energies are the same), and the initial speeds are the same
(so the kinetic energies are the same). When the balls hit the ground, the
total mechanical energy (potential + kinetic) for each ball is converted to
kinetic energy
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( 6.00 m s)2
( 2) ( 9.80 m
s2
)
= 1.84 m
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Conservation of Energy,
Example 2 (Pendulum)
„
„
Answer:
.
f
„
„
As the pendulum swings,
there is a continuous change
between potential and
kinetic energies
At A, the energy is potential
At B, all of the potential
energy at A is transformed
into kinetic energy
„ Let zero potential energy
be at B
At C, the kinetic energy has
been transformed back into
potential energy
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What if we want the ball
to swing at least half a
circle about the pole? How?
Answer:
l − 2r
cos Ө =
Ө
l: the length of the
string
r: radius of the circle
centered at the pole
l
Example. Jane, whose mass is
50.0 kg, needs to swing across a
river (having width D) filled with
man-eating crocodiles to save
Tarzan from danger. She must
swing into a wind exerting
constant horizontal force F, on a
vine having length L and initially
making an angle θ with the
vertical. Taking D = 50.0 m,
F = 110 N, L = 40.0 m, and
θ = 50.0°.
(a) With what minimum speed must Jane begin her swing in
order to just make it to the other side?
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(a)
0 potential level
m
110 N
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With what minimum speed
must Jane begin her swing in
order to just make it to the
other side?
(reverse swing ball)
Answer:
D = L sin θ + L sin φ
50.0 m = 40.0 m ( sin 50° + sin φ )
φ = 28.9°
(b) Once the rescue is complete, Tarzan and Jane must swing
back across the river. With what minimum speed must they
begin their swing? Assume that Tarzan has a mass of 80.0 kg.
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(b) Once the rescue is complete,
Tarzan and Jane must swing back
across the river. With what
minimum speed must they begin
their swing? Assume that Tarzan
has a mass of 80.0 kg.
Answer:
50 m
0 potential level
Jane
initial KE + initial PE +
workdone by the wind force
= final PE at destination
initial KE + initial PE –
workdone in overcoming the wind force
1 2
m vi + m g ( − L cosφ ) + FD ( +
=1) = 0 + m g( − L cosθ )
2
1
130 kg vi2 + 130 kg 9.8 m s2 ( −40 m cos28.9°) + 110 N ( 50 m
2
= 130 kg 9.8 m s2 ( −40 m cos50°)
= final PE at destination
- FD =
(
(
)
)
)
1
130 kg vi2 − 4.46 × 104 J+ 5 500 J= −3.28 × 104 J
2
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vi =
2( 6 340 J)
130 kg
= 9.87 m s
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Conservation of Energy,
Example 3 (Spring Gun)
The launching mechanism of a toy gun consists
of a spring of unknown constant k. When the
spring is compressed 0.12 m, the gun, when
fired vertically, is able to launch a 35 g projectile
to a maximum height of 20 m above the position
of the projectile before firing. Neglecting all
resistive forces, determine the spring constant k.
Answer:
Conservation of Energy,
Example 3 (Spring Gun)
h
Find the speed of the projectile as it
moves through the equilibrium position
of the spring (at XB = 0.12 m).
Answer:
½ kx2 = ½ mvB2 + mgxB
½ x (953 N/m) (0.12 m)2
= ½ x (0.035 kg)vB2 +
(0.035 kg)(9.8 m/s2)(0.12 m)
elastic potential energy
gravitational potential energy
½ kx2 = mgh
k = 2mgh / x2
= 2 (0.035 kg)(9.8 m/s2)(20 m) / (0.12 m)2
= 953 N/m
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Nonconservative Forces
„
„
A nonconservative force does not
satisfy the conditions of conservative
forces
Nonconservative forces acting in a
system cause a change in the
mechanical energy of the system
Recall that conservative forces have these two equivalent properties:
• The work done by a conservative force on a particle moving between
any two points is independent of the path taken by the particle.
• The work done by a conservative force on a particle moving through
any closed path is zero.
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vB = 19.7 m/s
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Mechanical Energy and
Nonconservative Forces
„
In general, if friction is acting in a
system:
„
„
„
„
ΔEmech = ΔK + ΔU = -ƒkd
Difference in signs is due to the loss and
gain in the energy
ΔU is the change in all forms of potential
energy
If friction is zero, this equation becomes
the same as Conservation of Mechanical
Energy, ΔEmech = 0, i.e., no change in
mechanical energy.
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Nonconservative Forces, cont
„
„
The work done against
friction is greater along
the brown path than
along the blue path
Because the work done
depends on the path,
friction is a
nonconservative force
Nonconservative Forces,
Example 1 (Slide)
ΔEmech
ΔEmech
ΔEmech
ΔEmech
= ΔK + ΔU
=(Kf – Ki) + (Uf – Ui)
= (Kf + Uf) – (Ki + Ui)
= ½ mvf2 – mgh
= -ƒkd
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We will first analyze the sequence of arrivals of these
metal balls on the tracks. Which metal ball will reach
the bottom first?
C
D
B
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Example. Mary and John are at a water park. There
are two water slides that start with the same height and
end at the same height. Slide A has more gradual slope
than slide B. John likes slide B better and he says he
reaches at a faster speed with slide B when touching
the water because he notes that he got to the bottom
level in less time on slide B than on slide A as measured
with his stop watch. Mary who does not carry a stop
watch says she touches the water with the same speed
on either slide. Who is correct and why? Both slides
have negligible friction.
(demo 5 tracks)
A
Answer:
C
Mary is correct. The velocity when they touch the
water is the same at either slide because the vertical
distance of decent is the same. mgh = 1 mv 2 .
D
gӨ
A
g sin Ө
g
Ө
B
2
B
A
g sin Ө
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John is partially correct. As the acceleration along the slide
(g sin Ө) is larger on B, the velocity is increased at a faster
rate on slide B during the earlier duration so it takes lesser
time (or is faster) to get to the bottom on slide B, but the
final speeds are the same on both slides.
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Example. A 5.00-kg block is set into motion up
an inclined plane with an initial speed of 8.00 m/s.
The block comes to rest after traveling 3.00 m
along the plane, which is inclined at an angle of
30.0° to the horizontal. For this motion determine
(a) the change in the block's kinetic energy,
(b) the change in the potential energy of the
block-Earth system, and (c) the friction force
exerted on the block (assumed to be constant).
(d) What is the coefficient of kinetic friction?
Answer:
(a)
(
Nonconservative Forces,
Example 2 (Spring-Mass)
30°
„
30°
mg
)
1
1
m v2f − vi2 = − m vi2 = −160 J
2
2
ΔU = m g( 3.00 m ) sin 30.0° = 73.5 J
ΔK =
(b)
(c) The mechanical energy converted to heat due to friction is
160 J – 73.5 J = 86.5 J. As W = f x d, f = W /d, so
(d)
f = μkn = μkm g cos30.0° = 28.8 N
μk =
28.8 N
( 5.00 kg) ( 9.80 m
)
s2 cos30.0°
„
86.5 J
f=
= 28.8 N
3.00 m
„
„
„
ΔEmech = -ƒkd
= 0.679
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Nonconservative Forces,
Example 3 (Connected Blocks)
„
Without friction, the
energy continues to be
transformed between
kinetic and elastic
potential energies and
the total energy
remains the same
If friction is present, the
energy decreases
The system consists of the
two blocks, the spring, and
Earth
Connected Blocks, cont
„
Gravitational and potential
energies are involved, and
friction is not 0
„
The kinetic energy is zero if
our initial and final
configurations are at rest
„
„
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Block 2 undergoes a
change in gravitational
potential energy
The spring undergoes a
change in elastic potential
energy
The coefficient of kinetic
friction can be measured
Why is the coefficient of
static friction not used?
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Example. A 1.00-kg object slides to the right on a
surface having a coefficient of kinetic friction 0.250. The
object has a speed of vi = 3.00 m/s when it makes
contact with a light spring that has a force constant of
50.0 N/m. The object comes to rest after the spring has
been compressed a distance d. The object is then forced
toward the left by the spring and continues to move in
that direction beyond the spring’s unstretched position.
Finally the object comes to rest a distance D to the left of
the unstretched spring. Find (a) the distance of
compression d, (b) the speed v at the unstretched
position when the object is moving to the left, and (c) the
distance D where the object comes to rest.
(b) the speed v at the unstretched position
when the object is moving to the left
(figure 4)
Answer:
Now the spring is at the natural position so the elastic
potential energy in the spring is zero. The remaining
energy left for pushing the block to the left is the
initial kinetic energy less the workdone by friction
where the block has travelled a distance of 2d.
2
Answer:
(a)
X m
Initial amount of energy in the system is
=
.
This energy is given to the elastic potential energy in
the spring when it is compressed and the workdone
by friction (figure 3), i.e.,
=
=
By the roots of quadratic
equation ax2 +bx + c = 0 Æ
+
+
f = μmg = 0.25 x 1 x 9.8
= 2.45 N
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(c) the distance D where the object comes to rest.
Energy Diagrams and Stable
Equilibrium
Answer:
As the spring is elastic, the potential energy
gained and subsequently returned by the
spring has a net 0 value (we assume that no
heat is generated in the spring or it is
negligible as the deformation is done only
once).
„
equilibrium
When the (detached) object comes to rest, the
initial kinetic energy is transformed to the work
done by friction travelled with a distance
of ( d
+
d
+
The x = 0 position is
one of stable
„
D).
=
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Configurations of
stable equilibrium
correspond to those
for which potential
energy U(x) is a
minimum
A ball in a valley is in
the state of stable
equilibrium.
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Energy Diagrams and Unstable
Equilibrium
„
„
„
„
Fx = 0 at x = 0, so the
„
particle is in equilibrium
For any other value of
x, the particle moves
away from the
equilibrium position
This is an example of
unstable equilibrium
Configurations of
unstable equilibrium
correspond to those for
which U(x) is a
maximum
Neutral Equilibrium
„
A ball on top of a hill is
in the state of unstable
equilibrium.
Neutral equilibrium occurs in a
configuration when U is constant over
some region
A small displacement from a position in
this region will produce neither
restoring nor disrupting forces
A ball on a flat ground
is in the state of neutral
equilibrium.
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Example. A particle moves
along a line where the
potential energy of its system
depends on its position r as
graphed in Figure. In the limit
as r increases without bound,
U(r) approaches +1 J.
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(a) Identify each
equilibrium position for
this particle. Indicate
whether each is a point of
stable, unstable or
neutral equilibrium.
Answer:
(a) Identify each equilibrium
position for this particle.
Indicate whether each is a point of stable, unstable or neutral
equilibrium.
There is an equilibrium point wherever tangent to the graph
of potential energy is horizontal:
At r = 1.5 mm and 3.2 mm, the equilibrium is stable.
Now suppose that the system (PE + KE) has energy −3 J.
Determine (b) the range of positions where the particle can
be found, (c) its maximum kinetic energy, (d) the location
where it has maximum kinetic energy.
At r = 2.3 mm, the equilibrium is unstable.
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A particle moving out toward r→ ∞ approaches neutral
equilibrium.
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Now suppose that the
system has energy −3 J.
Now suppose that the
system has energy −3 J.
Determine (b) the range of
positions where the particle can
be found
(d) Find the location where it has
maximum kinetic energy
Answer:
If the system energy is
(c) Find its maximum
–3 J and the system is
kinetic energy
isolated, its potential
energy must be less
Answer:
than or equal to –3 J.
K +U = E
Thus, the particle’s
K m ax = E − U m in = −3.0 J− ( −5.6 J) =
position is limited to
0.6 m m ≤ r ≤ 3.6 m m K m ax = E − U m in = −3.0 J− ( −5.6 J) = 2.6 J
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Answer:
Kinetic energy is a maximum
when the potential energy is a
minimum, at r= 1.5 m m
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